The Structure of Matricially Quasinormal Tuples

University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange University of Tennessee Honors Thesis Projects University of Tennessee Honors Program Spring 4-2006 The Structure of Matricially Quasinormal Tuples Katherine Collier Frederick University of Tennessee-Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_chanhonoproj Recommended Citation Frederick, Katherine Collier, "The Structure of Matricially Quasinormal Tuples " (2006). University of Tennessee Honors Thesis Projects. https://trace.tennessee.edu/utk_chanhonoproj/956 This is brought to you for free and open access by the University of Tennessee Honors Program at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in University of Tennessee Honors Thesis Projects by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contact trace@utk.edu.

Title: The Structure of Matricially Quasinormal Tuples Author: Katherine Collier Frederick Faculty Mentor and Department: Jim Gleason, Mathematics Date: April 12, 2006

Abstract Quasinormal operators are a class of operators that exists between normal operators, those that commute with their adjoint, and subnormal operators, normal operators restricted to an invariant subspace. Specifically, a quasinormal operator on an infinite-dimensional Hilbert space commutes with its adjoint times itself. In this paper we examine one possible generalization of the concept of quasinormal operators to the multivariable setting: matricially quasinormal tuples. To accomplish this purpose we follow the research of single variable matricially quasinormal tuples performed by John B. Conway and Pei Yuan Wu and expand their research to the more general multi variable setting. We study some basic properties of these commuting tuples and create a related structure theorem.

The Structure of Matricially Quasinormal Tuples Katherine Frederick and Jim Gleason April 11, 2006 Abstract In this paper we look at one possible generalization of the concept of quasinormal operators to the multivariable setting. We then study some basic properties of these commuting tuples and create a related structure theorem. 1 Introduction Quasinormal operators are a class of operators that exists between normal operators, those that commute with their adjoint, and subnormal operators, normal operators restricted to an invariant subspace. They include normal operators, and all quasi normal operators are subnormal. Specifically, a quasinormal operator, T, on an infinite-dimensional Hilbert space, H, commutes with its adjoint times itself, or T(T*T) = (T*T)T. The quasi normal operator's curious position between normal and subnormal operators makes quasinormal operators an interesting example of operators which are close to being normal, with the basic example being the unilateral shift, U+. The most useful structural theorem of a quasi normal operator, T, is that it is unitarily equivalent to an operator o ) A o where N is a normal operator and A is a positive self-adjoint operator. If T is pure, which means it has no reducing subspace on which it is normal, then the normal operator is unnecessary in the previous structural theorem [1]. A natural direction in which to continue researching an operator is to examine its properties in the multivariable case. Commuting normal tuples of operators act similarly to their single variable counterparts and provide no important surprises. Subnormal tuples of operators have also been well-studied. However, a multivariable generalization for quasinormal operators has received far less attention. Theorems regarding multivariable quasi normal tuples and 1

their structures give a useful glimpse at a transition class between subnormal and normal tuples of operators. Three types of multivariable quasinormal tuples have been introduced: matricially, jointly, and spherically quasinormal tuples [3]. The class closest to the normal tuples, and therefore most readily studied, is the matricially quasinormal tuples. Definition 1.1. A commuting tuple T = (T 1,.., Td) is called matricially quasinormal if Ti commutes with Tj*Tk for all i,j, k = 1,...,d. In this article, we follow the path of [2], and develop representation theorems for matricially quasinormal tuples. 2 Representation Theorems The first of the representations that we will examine is related to a matrix structure of the matricially quasinormal tuple. This representation was given in [3] as one possible generalization of quasinormal operators to the multivariable setting. Some notational explanation is necessary before we begin. For commuting tuple, T = (T 1,., Td), and multi-index, a = (al,"" ad) E Zi, we define ITI" = IT11'"... ltd I"d where ITi I = (T;*Ti) 1 and T" = Tf'... T:t d Proposition 2.1 ([3], Theorem 2.2). Let T = (T 1,.., Td) be a pure commuting tuple. Then Ti commutes with Tj*Tk for all i, j, k if and only ift;' [Tj*' T k ] = 0 for all i, j, k. In this case, there is a subspace.c and positive commuting operators Ai on.c such that 0 T,~ ) Ai 0 ( Ai 0 on H = EB~=l.c. Using this representation we can see that matricially quasinormal tuples are subnormal, and if they are pure, we can find the structure of its minimal normal extension. Proposition 2.2 ([3D. If T = (Tl,..., Td) is a pure matricially quasinormal tuple, then T is subnormal with normal extension, N = (N 1,..., Nd) of the form Ni=(Tt 0) C i Ti 1 on H EB H where Ci = ([T;*, TiD"2. Furthermore, N is the minimal normal extension of T if and only if T is pure. 2

Proof. Let T = (T 1,.., T d ) be a matricially quasi normal tuple. Let Ni=(T;* 0) Ci Ti 1 where C i = ([T;*, Til)"2. To check that each Ni is normal, examine the selfcommutator. [Nt, Nil = Nt Ni - NiN;* = (~i ~!) (~: ~i) - (~: ~i) (~i ~!) _ ( C; - [T;', Til CiTi - T;*Ci ) _ 0 - T;*Ci - CiTi [T;*, Til- C; - 1 1 1 since C i = ([T;*, TilP and CiTi = ([T;', Til) "2 Ti = T;* ([T;*, Til)"2 = T;'Ci. N is a commuting tuple since = (T;*T; 0) _ (TIT;' 0) = 0 CiT; + TiCj TiTj CjT;' + TjCi TjTi. Thus, since T is pure N = (Nr,..., N d ) is the minimum normal extension. D Since Ai commutes with Aj for all j, we see that if T is a matricially quasinormal tuple, then T = U+ 0 A where U+ is the unilateral shift of multiplicity 1 and A = (A 1,..., Ad) is a commuting tuple of positive operators. Definition 2.3. A tuple A = (A 1,..., Ad) on a Hilbert space 'H is called cyclic if there is a vector e in 'H such that the linear span of {A"'e : a E Zi} is dense in 'H. Proposition 2.4. A pure matricially quasinormal tuple T = (T l,..., T d ) on 'H is unitarily equivalent to U+ 0A, where A = (Al,...,Ad) is a positive cyclic tuple on a Hilbert space 12, if and only if there is a vector e in 'H such that'h is the closed linear span of {ITI"'Ti'3 e : a, (3 E Zi}. Proof. Assume there exists an e E 'H such that 'H = V {ITI"'Ti'3 e : a,(3 E Zi}. It is sufficient to say that 'H = 12(00) and Ti = U+ 0Ai. If such a vector e exists, let e = (eo, el,... ), with ei in L. Note that and each A 0 ITil = ( 0' Ai 3

For 1131 ~ 1, the zeroth coordinate of ITI""T,6e = ITII"'"... ltd I ""qf'... T:de is O. However, the zeroth coordinate of ITI""e is A""eo. Thus, eo is a cyclic vector for A. Now assume eo is a cyclic vector for A and let e = (eo, 0, 0,...) in [)co). The for every lal, 1131 ~ 0, every coordinate of ITI""T,6e is zero except for the (.II th d' t h' h' A"'" A""d A,6, A,6d A"'" +,6, A""d+,6d I fj - coor lila e, w IC IS I.., d I'.. d eo = I... d eo = A",,+,6eo. To show that A has dense range, fix j ~ 1. Let f E such that for all a E Zi 0= (f,a""(~f=ia{eo)) = (f,~f=ia""aieo) = ~f=1 (f,a""a{eo) = ~f=1 (f,a{a""eo) = ~f=1 (AU,A""eo) = (~f=iau,a""eo) As a result, ~f=i AU = and (~f=i AU'!) = 0. Thus, AU i = 1,...,d and f E n~=1 kerai = {O}. Thus, H is the closed linear span of {ITI""T,6e: a,j3 E Zi}. for all For a matricially quasi normal tuple T = (TI,..., Td), let ~(T) be the norm closed algebra generated by {TI,...,Td and {ITII,...,ITdl}. T is ~(T)-cyclic if there is a vector e such that {Re : R E ~(T)} is dense in H. By the previous proposition, T is ~(T)-cyclic if and only if A is cyclic. If for i = 1,...,d, 1/ is a positive Borel measure on [O,oo)d, i/ will denote the measure on Cd defined by d -( iii, iijd ) - 1 db db d ( ) I/rle,...,rde -(27f)d 1'" d I/r. This brings us to our main result. Theorem 2.5. If T = (TI'...,Td) is a pure matricially quasinormal tuple that is ~(T)-cyclic, then there is a measure 1/ on K = [O,IITIII] x [0, IIT211] x... x [O,IITdlll such that, if ~ is the closed linear span of {lzl""z,6 : a,j3 E Zi} in 2 (i/), then, for each i = 1,...,d, Ti is unitarily equivalent to multiplication by Zi on~. Proof. Assume that II T 11= 1, H = (00), and Ti = U+ Ai' T is ~(T)-cyclic if there is a vector e such that {Re: R E ~(T)} is dense in H. Since T is ~(T) cyclic, then A is a positive cyclic tuple. Since T is a matricially quasi normal tuple, each Ai is self-adjoint. Thus, nker Ai = 0, and there exists and measure 1/ on K such that there exists unitary U with Ai = U' MriU for all i. Also, note that 1/( {O}) = O. For all 13 E Zi, let ~,6 be the closed linear span of {I Z I "" z,6 : a E Zi} in C 2 (i/). Each ~,6 is orthogonal since D 4

= _1_ jr"+f3+'"y+t(eii31ih... eii3dod) (e-i'"yllh... e-i'"ydod)dvd(o)n?l = 0 (27r )d if (3 = T. For J E 2(V) and (3 E zt define 9 on Cd\O by g(z) = Izl-,6z,6 J(lzl). Since ii( {O}) = 0, this defines 9 a.e. Also, Define V,6 : 2(V) -t 2(ii) by (V,6f)(z) = Izl-,6z,6 J(lzl). Since 11 V,6J 112=11 J 11 2, V,6 is an isometry. Let (3 E zt be fixed. Let a,6 = {p E C[r] : p(r) = r,6q(r) for some q E qr]} where r = (rl'...,rd). Then by the Stone Weierstrass Theorem, any continuous function on K can be uniformly approximated by polynomials in a(3 which vanish at the origin. Thus, we have that a(3 is dense in 2(V). Let p E a,6, then V,6P E!J\,6. So V,6 : 2(V) -t!j\(3. On the other hand, if 9 E!J\(3, then there is a sequence of polynomials {Pn} such that J Pn(lzl)z(3 - g(z)2dii(z) -t 0 as n -t 00. But L Ir,6Pn(r) - r(3pm(rwdv(r) = j Iz(3Pn(z) - Z(3Pm(z)1 2 dii(z) -t O. So {r(3pn} is Cauchy in 2(V); let J be the limit of the sequence {r(3pn} in 2(v). But V,6(r(3Pn) = z(3pn; since V,6 is continuous, V(3(f) = g. So V(3 is an isometry of 2(V) onto!j\,6. For each (3, V(3 : 2(v) -t!j\(3 is unitary. Since 1i = EB(3EZ d = (00) + and V : 1i -t!j\ by V = EB,6EZd V(3. Let M = (Mz\? M Z2 ' ' MzJ on!j\. If + a,(3 E zt, then Izl"z(3 E!J\,6. V- 1 (lzl"z(3) = V(3-1(lzl"z(3) = ra+(3 E L2(vk Thus, (EB,6 EZi U)Ti (EB,6 EZi U*)V-l(lzl"z,6) = (EB(3 EZ i U)Ti (EB,6E Z i U*)(r,,+(3) = Mri (r"+,6) = r"+,6 ri E L 2 (v),6+ei. Hence, V(EBU)Ti(EBU*)V*(l z l"z(3) = V(EBU)Ti(EBU*)(r,,+(3) = V(r"+(3. ri) Thus, MZi = V(EBU)Ti(EBU*)V*. = Izl"z,6+e i = MZi (l z l"z(3). o 5

References [1] Arlen Brown. On a class of operators. Proc. Amer. Math. Soc., 4:723-728, 1953. [2] John B. Conway and Pei Yuan Wu. The structure of quasinormal operators and the double commutant property. Trans. Amer. Math. Soc., 270(2):641-657, 1982. [3] Jim Gleason. Quasinormality and commuting tuples. preprint. 6